In classification, how does the number of classes affect the model size and amount of data needed to train?
The most obvious way more classes increase the network size it the output layer, but I don't believe there is a rule of thumb for the size of the entire network.
As I understand it, there is no clear answer how big a network needs to be to achieve a certain performance with regard to the number of layers compared to the number of classes. This is a very active research field, just as an example compare the size of EfficientNet to other State of the art models when it was introduced and you can see the size difference.
Regarding the data needed, in The Deep Learning Book (https://www.deeplearningbook.org/) (which is a few years old now) they state as a general that with the models available at that time, you needed ~5000 examples per label for acceptable performance, while to exceed human performance (their words) you would need abut 10 million labeled examples.